Okay, now I have a revision to my question...
I found someone else in the RB website forum who had a similar problem with a song he had tried to download and gotten an error while downloading. That DID happen when I was trying to download (I have a crappy internet connection) so I tried again today and while it worked, it doesn't show up anywhere.
I tried again with another song and when it downloaded on the first try, it automatically copied to the SD and now shows up in the song list(s). So, apparently the issue only happens if you fail to download the song on the first try.